Can anyone help me with this proof? Am I on the right track?
Let A and B be equivalent square matrices. Prove that A is nonsingular if and only if B is nonsingular

Given: Matrix A and Matrix B are equivalent square matrices.
Since we are given a biconditional statement, we need to prove two situations:
If Matrix A is nonsingular then Matrix B is nonsingular
If Matrix B is nonsingular then Matrix A is nonsingular

Alright. Suppose A is nonsingular, and \(A=PBQ\) for some non-singular matrices P,Q. This means that A, P, Q are invertible. Then \[B=P^{-1}AQ^{-1}\]But since \(A^{-1}\) exists, look at the product\[P^{-1}AQ^{-1}QA^{-1}P =BQA^{-1}P=\text{Id}\]So B has an inverse and must be non-singular.
This is how I would do it. Just switch A and B to prove the other statement you need.

That's the right idea. We're finding an \(X\) such that \(AX=\text{Id}\). So if \[AX=P^{-1}BQ^{-1}X=\text{Id},\]you need to find \(X\) in terms of \(P,Q,\) and \(B\) (and/or their inverses).
Make more sense?

Ok. I may have got it.
Suppose B is nonsingular and B=PAQ for some non-singular matrices P and Q. That means B, P, and Q are invertible.
|dw:1361526378870:dw|
|dw:1361526405075:dw|
|dw:1361526433824:dw|
|dw:1361526465398:dw|
Since B^-1 exist we look at the product of nonsingular matrices.
|dw:1361526502682:dw|
Therefore A has an inverse and it must be nonsingular

Given: Matrix A and Matrix B are equivalent square matrices. Since we are given a biconditional statement, we need to prove two situations: If Matrix A is nonsingular then Matrix B is nonsingular If Matrix B is nonsingular then Matrix A is nonsingular

you don't have to go through the rigorous way like you did the first one, Both P^-1 and Q^-1 are invertible matrices. If you take P and Q on the side of B, you will get same expression as you did on the first one.